The radius of analyticity of periodic analytic functions can be characterized by the decay of their Fourier coefficients. This observation has led to the use of so-called Gevrey norms as a simple way of estimating the time evolution of the spatial radius of analyticity of solutions to parabolic as well as non-parabolic partial differential equations. In this paper we demonstrate, using a simple, explicitly solvable model equation, that estimates on the radius of analyticity obtained by the usual Gevrey class approach do not scale optimally across a family of solutions, nor do they scale optimally as a function of the physical parameters of the equation. We attribute the observed lack of sharpness to a specific embedding inequality, and give a modified definition of the Gevrey norms which is shown to finally yield a sharp estimate on the radius of analyticity.
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